A fun Fibonacci number surprise with a 1, 2, Sqrt[5] right triangle

My son had a little trouble with this problem from the 2009 AMC 10 B yesterday:

I should him how to solve the problem a few different ways – including by folding!

One of the ways I talked about was finding a rough answer by approximating $\sqrt{5}$ by the fraction 9/4. We found this number by looking at a calculator and seeing that $\sqrt{5} \approx 2.23607.$

As I thought about the problem more last night, I realized that 9/4 is part of the continued fraction approximation for $\sqrt{5}$. The first couple of approximations that you find using continued fractions are:

2, 9/4, 38/17, and 161/72.

If I approximate the hypotenuse of the original right triangle with those numbers, I get the following approximations for the length of BD, which are all ratios of consecutive Fibonacci numbers:

If you are familiar with continued fractions and especially the continued fraction approximations for the golden ratio, the emergence of the Fibonacci numbers probably isn’t a huge surprise. I missed it the first time, though, and think that students might really enjoy seeing this little Fibonacci surprise.

Sharing John Cook’s Fibonacci / Prime post with kids

Saw a neat post from John Cook about using a fun fact about the Fibonacci numbers to prove there are an infinite number of primes:

Infinite Primes via Fibonacci numbers by John Cook

Funny enough, we’ve played with the Fibonacci idea before thanks to Dave Radcliffe:

Dave Radcliffe’s Amazing Fibonacci GCD post

That project was way too long ago for the kids to remember, so today we started by just trying to understand what the Fibonacci identity means via a few examples:

Next we looked at the idea from Cook’s post that we need to understand to use the Fibonacci identity to prove that there are an infinite number of primes. The ideas are a little subtle, but I think the are accessible to kids with some short explanation:

We got hung up on one of the subtle points in the proof (that is pointed out in the first comment on Cook’s post). The idea is that we need to find a few extra prime numbers from the Fibonacci sequence since the 2nd Fibonacci number is 1. Again, this is a fairly subtle point, but I thought it was worth trying to work through it so that the boys understood the point.

Finally, we went upstairs to the computer to explore some of the results a bit more using Mathematica. Luckily Mathematica has both a Fibonacci[] function and a Prime[] function, so the computer exploration was fairly easy.

One thing that was nice here was that my older son was pretty focused on the idea that we might see different prime numbers in the Fibonacci list than we saw in the list of the first n primes. We saw quickly that his idea was, indeed, correct.

This project made me really happy 🙂 If you are willing to take the Fibonacci GCD property for granted, Cook’s blog post is a great way to introduce kids to some of the basic ideas you need in mathematical proofs.

An infinite descent problem with pentagons

The boys had a nice hiking trip to Mt. Osceola today.

When they got back we did a little follow up to yesterday’s project on infinite descent:

Infinite Descent with kids inspired by Jim Propp

That project came from reading the lastest Jim Propp blog post:

Fermat’s Last Theorem: The Curious Incident of the Boasting Frenchman

So, the kids were a little tired tonight, but we still managed to have nice (and short) project tonight discussing this problem from Propp’s blog:

“If you liked the proof that there is no integer-sided golden rectangle, you might try to prove, by similar means, that there is no regular pentagon whose five equal sides and five equal diagonals all have integer length.”

The Zometool set makes this project especially fun!

However, because I am also tired, tonight’s write up will be short (and not edited – I’m going to bed).

In part 1 we built some pentagons and tried to see if we could see a pattern in the lengths of the sides and the diagonals. I’d left the length of the sides of the larges pentagon as a mystery to help motivate finding the pattern:

The boys didn’t see the pattern so we moved to the whiteboard – when you write the lengths down, the patter is much easier to see:

So, next we moved back to the couch to test if the pattern we found worked – it did! My older son wanted say that the longs were an integer and that idea threw me off in our discussion here. To try to get things back on a more correct path (meaning, one that was more clear to me given that I was tired 🙂 ), we went back to the witeboard after this part of the discussion:

ok – so back to the white board to see if the boys could tackle the infinite descent proof. They had the idea for how to start the proof in the last video – assume that we’ve found a regular pentagon with integer sides and diagonals. How do we find the smaller pentagons now?

It took a few minutes, but we got there 🙂

It is fun seeing kids struggle through a puzzle like this one, and the Zome set is almost a miracle helper for this particular problem. Fun little post hiking project 🙂

Infinite descent with kids inspired by Jim Propp

I saw another neat post from Jim Propp a few days ago:

Fermat’s Last Theorem: The Curious Incident of the Boasting Frenchman

The part that caught my attention as something that kids might find fun was the discussion of “infinite descent.” It reminded me a little of a Zometool project that we did last year:

Fibonacci Spirals and Pentagons with our Zometool set

We sure had a lot of Zome fun in the new house before we got furniture 🙂

To get going with the “infinite descent” discussion today I had the boys build another Fibonacci spiral shape from the Zometool set. We talked about the shape and my younger son remembered the connection to the Golden Rectangle, which was a nice surprise.

This shape is actually really fun to build because you don’t really get a sense for how quickly the shape grows until you try to fit it in your living room 🙂

My son remembering a connection with the Fibonacci Spiral and the Golden Rectangle made for a nice transition to the next part of the project. The first thing we needed to do was talk about what it means to be “similar” in geometry.

For the last part of the project we stepped through the infinite descent argument showing that the Golden Ratio cannot be the ratio of two integers.

The argument requires a little bit of algebra so I went through it slowly to make sure that my younger son could follow it. Hopefully he was able to follow it – and I think he did because at the end he asked about non-integer values for the ratio.

Pretty fun to start the project with a spiral that grows larger and larger and end with a spiral that shrinks and shrinks!

Another fun project inspired by Jim Propp’s blog. I originally wanted to look at the pentagon question mentioned in Propp’s post, but I think we’ll explore that question next weekend.

A fun counting excercise for kids suggested by Jim Propp

[note: sorry for the quick and unedited write up on this one. We are a little pressed for time today.]

Got a neat comment from Jim Propp on one of the Aztec diamond posts last night:

“There are other fun problems about counting tilings. The first one I’d give them is counting domino tilings of a 2-by-n rectangle. Then rhombus tilings of an equiangular hexagon with sides of length a,b,1,a,b,1. If the kids like these, let me know; I can suggest some fun follow-ups.”

I thought about the 2 by N rectangle tiling on the way home last night and thought it would, indeed, make a great project to do with the boys today.  Despite a little clumsiness keeping the domino tiles on the screen, it was really fun.

I started by introducing the problem and asking the boys to come up with a strategy to solve it. They had some good thoughts:

Following the strategy they came up with in the first part, we started by looking at some simple cases – 2×1, 2×2, 2×3, and 2×4. These cases allowed us to make a few guesses at the pattern:

In the prior video the kids guessed the tiling patter we were seeing 1, 2, 3, 5 was the Fibonacci numbers. Here we looked at the different ways to tile a 2×5 rectangle to see if there were 8.

Sorry we got a little clumsy with the tiles here – we needed a bit more room.

Now that we were feeling a bit more confident that we were seeing the Fibonacci numbers, the question was . . . why? How could we see the Fibonacci pattern in these tilings?

To understand what was going on, we looked to see if we could find a relationship between the combined tilings of the 2×2 and 2×3 recangles and the tilings of 2×4 rectangle. The relationship wasn’t obvious right away, but they did find it!

Sorry again for the clumsiness with the tiles.

Finally, as a last way of double checking the Fibonacci relationship, we looked to see if the tilings of the 2×5 rectangle related to the combined tilings of the 2×3 and 2×4 rectangles.

So, although we didn’t really do a complete formal proof of the relationship between Fibonacci numbers and the tiling patterns, we did hit on the main ideas in that proof. This activity is a great way to introduce some “mathematical thinking” ideas to kids. Thanks to Jim Propp for suggesting it!

Following up on Matt Enlow’s Fibonacci problem

About a week ago we took a quick look at a problem that Matt Enlow had posted on twitter:

Matt Enlow’s Fibonacci Problem

We had a little bit of extra time this morning, so I decided to revisit the problem to talk a little bit about modular arithmetic. I also really like this problem as an introductory proof problem, too, but that’ll have to wait for another day.

Also, sorry for writing the problem backwards at the start of the video, we manage to straighten it out once we look at the Fibonacci numbers mod 8.

Once we looked at the numbers mod 8, it was time to look at them mod 9 and see if we saw a pattern. I’d like to revisit this project some time to talk about ideas like why 8 = -1 mod 9.

So, I think this is a great problem for kids. It asks about a property that is fairly easy to understand and also provides a nice opportunity to introduce modular arithmetic. Lots of opportunities here to have some fun math conversations 🙂

Matt Enlow’s Fibonacci problem

Saw this tweet from Matt Enlow yesterday:

I liked this problem both as an illustration of mathematical thinking and as a problem you can share with kids.

For mathematical thinking, the point is made really well in this interview with Julie Rehmeyer. The ~5 min part beginning around 31:30 about proving that 0 + 0 = 0 is what I’m thinking of specifically:

Julie Rehmeyer’s “Inspired by Math” interview

I had a similar feeling on seeing Matt’s problem – it wasn’t obvious to me why the Fibonacci numbers should have this property, but I had some ideas about how you’d prove that they did:

For sharing with kids, I like this problem because it is (i) accessible, but (ii) probably not as obvious how to solve it. I shared it with my son this morning and although we didn’t solve the problem it was very interesting to hear the ideas he had about how you might go about solving it:

I’m excited to finish up this problem with my son later this week and also excited to try out this problem with a larger group of kids sometime.

Fibonacci spirals and Pentagons with our Zometool set

I picked another project from our Zome Geometry book for our Family Math project today. The topic that caught my eye last night was the Fiboancci spiral from Unit 7. We are still trying to figure out furniture for our living room, so we’ve got a great empty space for project like this one!

We started with a brief description of the project – interrupted by a few hiccups and the cat:

We made the first few squares and broke for a quick video. We used yarn to trace out the spiral. It was a little difficult to make the yarn bend into the proper spiral, so we just had it go straight across the squares. The kids noticed that the green Zometool struts would have accomplished the same goal. They were also surprised by how large the squares were getting.

We made it up to a Zome square with 21 long struts on a side and had a fun little discussion about the shape. My older son wondered how many more steps we could have gone if we used smalls instead of large blue struts. Turned out that it as many more squares as you’d think.

Also my younger son notice that the length of the yarn was also a Fibonacci pattern, that was a nice surprise.

Next up was a project involving pentagons. The connection to the prior project isn’t quite obvious in the beginning. The first step was a quick description of the project and a little bit of geometry:

After building a couple of steps in this project, my younger son noticed a Fibonacci-like pattern in the number of struts we were using. The interesting thing was that he saw the patter in the number only, but not in the size of the struts. Because neither kid initially saw the connection to the side lengths, it took a little bit of time to see the connection between steps, but eventually they found there way there.

We didn’t go all the way to filling the living room floor with this one, but we did build a few more stages. Talking through this shape this time, the kids notice a few more Fibonacci patterns. They were surprised to see the Fibonacci numbers show up talking about pentagons.

So, definitely a fun little project. The Fibonacci spiral piece is a nice project all by itself and it gives you a nice sense of how quickly the spiral grows. The Pentagon project doesn’t seem to be connected to the spiral project, but the Fibonacci numbers show up in an unexpected way. Fun morning.

Volume, Scaling, and a surprising relationship to the Fibonacci numbers.

We used our Zome Geometry book for another Family Math project today. The project we stumbled on used the blue struts to talk about volume and scaling. It starts by asking a seemingly simple question about the volume of two boxes you can make with the blue struts. Here’s the problem:

and the constructed boxes – I liked hearing their ideas about the volume of the larger box:

Next they re-built the “golden box” using only medium and short blue struts. Building the box this way gave them a little bit of extra insight into the volume. Their first guess about the volume was that it would be the same as a cube with side length equal to 2x the small blue struts.

We took a few minutes to build the cube they wondered about in the last video and looked to see if the volume of that cube matched the volume of the golden box:

Once we saw that the 2x2x2 cube was too large, the kids thought to build a cube out of the medium struts. It seems possible that these two structures have the same volume – but we need to find a relationship between the mediums and the small blue struts in order to calculate the volume.

While we had the camera off, the boys played with the medium and small struts to find an aproximate relationship. They used this approximation to estimate the volume of the golden box. It was really interesting to me to hear and see their approach here:

After we finished the project, my younger son was playing around with the struts a little more. I thought it would be fun to see if they could see the interesting relationship between the medium and small blue struts. We did a little postscript. Here they see the numbers 5, 8, 13 for the long struts and 3, 5, 8 for the short ones and guess that the next numbers will be 13 and 20.

We checked if 20 shorts matched 13 mediums and found a fun surprise!

So, a fun project with a surprise postscript. Who would have thought you’d see the Fibonacci numbers pop up just by building Zometool boxes 🙂

Dave Radcliffe’s amazing Fibonacci / GCD post

This morning I received a neat heads up on a really cool post from Dave Radcliffe on twitter:

I’ve been covering the concept of greatest common divisor with my younger son this week, so Dave’s post had some great timing.  It is such a neat formula that I thought I’d use it for a little talk with my kids tonight instead of doing our usual weekend Family Math.

We started by talking through Dave’s formula and writing down the first 15 Fibonacci numbers.  I then worked through a simple example to make sure that the kids understood the language of the formula:

next I had my younger son see if he could work through the formula using the 8th and 12th Fibonacci numbers.  He’s just learned the Euclidean algorithm, so I was assuming that he’d use that approach to find the greatest common divisor.  This formula turns out to be a great way to practice the Euclidean algorithm.

I wanted my older son to work through an example, too, and picked one with slightly larger numbers – the 10th and 15th Fibonacci numbers.  I have covered a bit more number theory with him and suspected that he would remember that you could find the greatest common divisor of two numbers by factoring.  I was glad that he did since it was a nice example for my younger son to see:

Finally, to do a few more complicated examples we went to the computer.  Since Mathematica has built in functions to calculate the GCD of any two numbers as well as to calculate any Fibonacci number, we could easily work with much larger numbers.  We did a couple of examples, including one where we forced a large greatest common divisor.

I’ve given no thought at all to how to prove Dave’s formula, but I do have a little bit of driving to do this weekend so at least I’ll have something to think about!  The boys seem to enjoy seeing fun little math facts like these, so despite not having a lot of context around this amazing formula I’m happy with how this project went.  Definitely not your standard greatest common divisor exercise!